Going down to 10 men in a game clearly cannot, of itself, be an advantage. That's one man less on the field to whom his playmates can use to unlock the other side's defense.
What, however, does appear to happen, sometimes, is that the team going down to 10 men is usually forced to become very defensive, so that teams that like to use pace and counter-attacking football lose their primary weapon of attack, and with no Plan B can easily become frustrated. Also, if the team that goes down to 10 men is, itself, a pacey counter-attacking side, they may find (ironically) that spaces open up for them as the opposition become over-confident and push forward more often.
There is no "advantage," as such, in going down to 10 men, it's more the case that managers are not as astute as they, perhaps, ought to be when re-organising the tactics of playing with or against 10 men.